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Theorem plymul0or 22969
Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )

Proof of Theorem plymul0or
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dgrcl 22922 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 22922 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10872 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 475 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 9620 . . . . . . 7  |-  0  e.  _V
65fvconst2 6107 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 17 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5849 . . . . . . . 8  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  (coeff `  0p ) )
9 coe0 22945 . . . . . . . 8  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2459 . . . . . . 7  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5851 . . . . . 6  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( NN0  X.  { 0 } ) `  (
(deg `  F )  +  (deg `  G )
) ) )
1211eqeq1d 2404 . . . . 5  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( NN0  X.  { 0 } ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  0 ) )
137, 12syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 ) )
14 eqid 2402 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2402 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2402 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2402 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 22943 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( (coeff `  F ) `  (deg `  F )
)  x.  ( (coeff `  G ) `  (deg `  G ) ) ) )
1918eqeq1d 2404 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  x.  ( (coeff `  G
) `  (deg `  G
) ) )  =  0 ) )
2014coef3 22921 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
2321, 22ffvelrnd 6010 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2415coef3 22921 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2524adantl 464 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
262adantl 464 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
2725, 26ffvelrnd 6010 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2823, 27mul0ord 10240 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( (coeff `  F
) `  (deg `  F
) )  x.  (
(coeff `  G ) `  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
2919, 28bitrd 253 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3013, 29sylibd 214 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3116, 14dgreq0 22954 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3231adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3317, 15dgreq0 22954 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3433adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3532, 34orbi12d 708 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  <->  ( (
(coeff `  F ) `  (deg `  F )
)  =  0  \/  ( (coeff `  G
) `  (deg `  G
) )  =  0 ) ) )
3630, 35sylibrd 234 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( F  =  0p  \/  G  =  0p ) ) )
37 cnex 9603 . . . . . . 7  |-  CC  e.  _V
3837a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
39 plyf 22887 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4039adantl 464 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
41 0cnd 9619 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
42 mul02 9792 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4342adantl 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4438, 40, 41, 41, 43caofid2 6553 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  oF  x.  G )  =  ( CC  X.  { 0 } ) )
45 id 22 . . . . . . . 8  |-  ( F  =  0p  ->  F  =  0p
)
46 df-0p 22369 . . . . . . . 8  |-  0p  =  ( CC  X.  { 0 } )
4745, 46syl6eq 2459 . . . . . . 7  |-  ( F  =  0p  ->  F  =  ( CC  X.  { 0 } ) )
4847oveq1d 6293 . . . . . 6  |-  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( ( CC  X.  { 0 } )  oF  x.  G
) )
4948eqeq1d 2404 . . . . 5  |-  ( F  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5044, 49syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
51 plyf 22887 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5251adantr 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
53 mul01 9793 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5453adantl 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5538, 52, 41, 41, 54caofid1 6552 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
56 id 22 . . . . . . . 8  |-  ( G  =  0p  ->  G  =  0p
)
5756, 46syl6eq 2459 . . . . . . 7  |-  ( G  =  0p  ->  G  =  ( CC  X.  { 0 } ) )
5857oveq2d 6294 . . . . . 6  |-  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( F  oF  x.  ( CC  X.  { 0 } ) ) )
5958eqeq1d 2404 . . . . 5  |-  ( G  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6055, 59syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
6150, 60jaod 378 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6246eqeq2i 2420 . . 3  |-  ( ( F  oF  x.  G )  =  0p  <->  ( F  oF  x.  G )  =  ( CC  X.  { 0 } ) )
6361, 62syl6ibr 227 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  0p ) )
6436, 63impbid 190 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   {csn 3972    X. cxp 4821   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519   CCcc 9520   0cc0 9522    + caddc 9525    x. cmul 9527   NN0cn0 10836   0pc0p 22368  Polycply 22873  coeffccoe 22875  degcdgr 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880
This theorem is referenced by:  plydiveu  22986  quotcan  22997  vieta1lem1  22998  vieta1lem2  22999
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